One of the most important brain functions supporting vision is the control of gaze, or eye and head movements. This project is devoted to understanding the nature of neuronal interactions underlying the control of gaze. Neurophysiological techniques can observe the responses of some neurons, but they can not reveal directly the nature of their functional interactions. This project constructs and tests mathematical models of sensory and motor functions involved in the control of gaze based on experimentally observed neuronal activity. In prior models of saccades (rapid, voluntary eye movements), the key role of controlling the movement's goal and speed was given to the superior colliculus (SC). The role of the cerebellum (C), in contrast, was assumed to involve the long-term regulation of saccadic accuracy. Analysis of neuronal responses from the SC and the C has led us to postulate a new model of how the brain controls visually guided saccades. The new model has two branches, one through the SC and one through the C, operating in parallel. This helps explain one of the earliest lesion studies in SC: even after bilateral SC ablations, the brain can still make saccades. Under normal conditions, the model uses the SC to control saccade beginnings, and the C to control saccade endings. More importantly, it lets us form a new interpretation of the role of each area. Thus, the SC is now believed to be generating a signal, in retinotopic coordinates, that represents where the selected target is. Thus, it is a sensory, and not a motor signal (as has been previously thought). Several recent studies have lent further support to this reinterpretation. The model also suggests a new, dual role for the cerebellum. The first, and more basic, role is to update the distribution of output activity during a movement. This is the role that corresponds to feedback in a classical model. The second, and more subtle, role is to recognize the constellation of inputs (i.e., sensory, motor, behavioral) we call the context before the movement. This context causes the C to inhibit activity at a specific locus in the cerebellar vermis. This in turns disinhibits activity on specific fastigial nuclear cells, which in combination with the first role of the cerebellum drives and steers the eye to its final position. These two roles lead us to describe the cerebellum as having a "pilot map", which controls the eye movement.